# HSC +2 Maths - Sample 5 Mark Questions - Part3

For Plus2 Maths Part1 and Part2 Important questions , watch these pages.

1) Examine the consistency of the system of equations:

2x + 3y + 7z = 5,

3x + y – 3z = 13,

2x + 19y – 47z = 32.

2) Find the adjoint of the matrix $\begin{bmatrix} 1\;\;\;\;\;\;\; 2 \\ 3\;\;\; -5 \end{bmatrix}$ and verify the result A(adj A) = (adj A) A = |A|.I

3)       (1) For any vector $\overrightarrow{r}$ prove that $\overrightarrow{r}=(\overrightarrow{r}.\overrightarrow{i})\overrightarrow{i}+(\overrightarrow{r}.\overrightarrow{j})\overrightarrow{j}+(\overrightarrow{r}.\overrightarrow{k})\overrightarrow{k}$

(2) Find the projection of the vector $7\overrightarrow{i}+\overrightarrow{j}-4\overrightarrow{k}$ on $2\overrightarrow{i}+6\overrightarrow{j}+3\overrightarrow{k}$.

4) Find the vector and Cartesian equation of the sphere on the join of the points A and B having position vectors $2\overrightarrow{i}+6\overrightarrow{j}-7\overrightarrow{k}$ and $-2\overrightarrow{i}+4\overrightarrow{j}-3\overrightarrow{k}$ respectively as a diameter. Find also the centre and radius of the sphere.

5) If P represents the variable complex number z, find the locus of P, if |2z-1| = |z-2|.

6) The headlight of a motor vehicle is a parabola reflector of diameter 12 cm and depth 4 cm. Find the position of bulb on the axis of the reflector for effective functioning of the head.

7) Obtain the Maclaurin’s series for $log_{e}(1+x)$.

8) Verify Lagrange’s law of mean for the function $f(x)=x^{3}-5x^{2}-3x$ on [1, 3].

9) Given $w=\frac{x}{x^{2}+y^{2}}$, where x=cos t; y=sin t; Find $\frac{\mathrm{dw} }{\mathrm{d} t}$

10) Evaluate: $\int_{0}^{2\pi}sin^{9}\frac{x}{4} \; dx$

11) Solve: $x\;dy=(y+4x^{5}\;e^{x^{4}}) \;dx$

12) G is a group, $a, b \;\varepsilon \;G$. Prove that $(a*b)^{-1}=b^{-1}*a^{-1}$

13) Construct the truth table for $(p \; \wedge \;q) \;\vee \;(\sim r)$.

14) The overall percentage of passes in a certain examination is 80. If 6 candidates appear in the examination, what is the probability that at least 5 pass in examination?

15)      (1) State and prove cancellation laws on groups.
(or)
(2) Four coins are tossed simultaneously. What is the probability of getting (i) exactly 2 heads; (ii) at least 2 heads; (iii) almost 2 heads?

16) Find the shortest distance between the parallel lines $\overrightarrow{r}=(i-j)+t(2i-j+k)$ and $\overrightarrow{r}=(2i+j+k)+s(2i-j+k)$.

17) If n is a positive integer prove that $(\sqrt3+i)^{n}+(\sqrt3-i)^{n}=2^{n+1}\;cos(\frac{n\pi}{6})$

18) Find the equation to the two tangents that can be drawn from the point (1, 2) to the hyperbola $2x^{2}-3y^{2}=6$.

19) Forces 2i+7j, 2i-5j+6k, -i+2j-k act at a point P whose position vector is 4i-3j-2k. Find the moment of the resultant of three forces acting at P about the point Q whose position vector is 6i+j-3k.

20) Prove that $sin\;x<x<tan\;x, x\epsilon (0,\frac{\pi}{2})$